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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5829_Библиотеки_им_академика_М_И_Перельмана.pdf
Real-Time 4D Cardiac Segmentation by Active Geometric Functions 229
Fig. 1 2D (line interface) and 3D (spherical interface) examples of interface representation using
explicit representation, level-set representation, and AGF representation for the same interface
To better shows the differences of explicit representation, level-set representation, and AGF representation, both 2D and 3D examples are shown in Fig.1.
In 2D, to represent a straight line with slope = 1 and passing through the origin
(0, 0), there can be three different representations:
1. Explicit representation: such as {...(−1, −1), (0, 0), (1, 1), ...}
2. Level-set: suchas the zerolevel-set of
Φ
(x, y, t)=(x −y)/√2, noteφis a signed
distance function, and it is defined on whole x–y plane instead of just on boundary
3. Surface function: such as y = x
Similarly in 3D, a unit spherical interface can be represented by an explicit mesh,
with corresponding level-set function (Fig. 1b) as a signed distance function is
1
Φ
(x, y, z)=
representation (Fig. 1c) is f (r,
√
(x2+ y2+ z2−1), and the corresponding geometric function
3
θ,ϕ
)=r −1 = 0.
Note since these three ways of surface representation are representing the
same surface, there are some similarities between the corresponding representation
functions for each scheme. It is obvious that both level-set functions and the
geometric functions have the same roots, with the fact that the coordinates used
in the explicit representation are digitized version of these roots. And geometrically,
these roots form the surface we are trying to represent. In other words, explicit
representations, level-set distance functions, and surface functions are just three
different equivalent forms of the actual interface function f for the targeting
surface. The 3D example also illustrates that non-Cartesian coordinate systems can
be used in geometric function representation to efficiently represent the desired
surfaces.
230 Q. Duan et al.
In this example, however, besides similarity and equivalence, it is more interesting to notice their differences. For example in 2D, the explicit representation is a
set of coordinates defined on the 2D x–y plain; the level-set function is an R
2
→ R
distance function defined on the whole 2D x–y plane; the surface function is an
R → R function defined only on a 1D x-axis. As a 1D function, surface function
representation has the advantage in efficiency compared with the other two common
representations. From the example, we can see that interface representation based
on surface function is more accurate than the explicit representation, and has less
dimensionality than both other methods. In this way, a surface can be efficiently and
accurately represented,which simplifies the downstream mathematical computation
such as energy minimizations during image segmentation. A more detailed analysis
of the AGF efficiency will come in a later section.
2.3 Driving Force
In general, as a deformable model framework, AGF can utilize any driving forces
used in other deformable model framework. In our particular example, we adopt a
variational framework in deriving the driving forces given its generic formulization
and flexibility in extension for additional energy term that is not covered in
our example. For example, we can use the Mumford–Shah segmentation energy
functional:
E(F,−→C )=
β
(F −G)2dV +
Ω
\−→C
α
Ω
\−→C
|∇F|2dV +
-
γ
ds, (3)
−→
C
in which−→C denotes the smoothed and closed segmented interface, G represents
the observed image data, F is a piecewise smoothed approximation of G with
discontinuities only along−→C ,and
Ω
denotes the image domain. Given the flexibility
of variational frameworks, other segmentation energy functionals may also be easily
adopted.
For the segmentation of an N-dimensional image dataset, the active geometric
functions framework will solve an (N −1)-dimensional variational problem, explicit
representation will solve an N-dimensional problem, and the level-set framework
will solve an (N + 1)-dimensional variational problem. It is obvious that AGF
framework has advantages in dimensionality reduction when compared with the
other two deformable model formulations, at the cost of some flexibility, given the
assumption of 1 versus (N −1) coordinate mapping. However,such surfacemapping
can be further modified via the combination with finite element models. In addition,
for typical medical applications, biological surfaces are relatively smooth and well
represented with relatively simple geometric functions.
Real-Time 4D Cardiac Segmentation by Active Geometric Functions 231
2.4 Efficiency of AGF
The efficiency benefit of AGFs comes from three aspects: a dimension-reduced
surface representation, use of efficient function basis, and dimension reduction
of optimization problem. The first two aspects can greatly reduce the number of
parameters used in the optimization procedure.
When comparing the efficiency of different deformable model frameworks, one
important factor has to be paid attention to: the number of parameters needed to
present the same interface is different for the three methods. Level-set uses every
grid point (within the narrowband); explicit parametric models use point lists on
the interface, assuming linear connectivity, which usually is less dense than grid
points; AGF uses function basis to represent the interface, which extremely reduces
the number of parameters. For example, in AGF, only one parameter is needed to
represent a circle in any size, whereas the number of parameters needed for level-set
and parametric models increases exponentially with the radii of the circle.
For example in 2D, to represent a 100-pixel diameter circle, SFA uses one
parameter by using a Fourier basis, a parametric model could use 63-node points
(π×d/5) assuming five-pixel spacing (with 63×2 = 126 parameter), and level-set
could use 942 (3 ×π×d) grid points assuming a minimal narrowband used, i.e.,
three-pixel narrowband (with 942 ×2 = 1,884 parameters). Here, computational
complexity for level-set with narrowband is linear with the parametric model since
the dimensionality N = 2anda
For 3D, it is totally a different story. To present a 100-voxel diameter sphere,
SFA uses one parameter by using a Fourier basis, a parametric model could use
1,257 nodes (π×d
2
/(5 ×5)) assuming five-pixel spacing (with 1,257 ×3 = 3,771
parameters), and level-set could use 282,743 (3 ×3 ×π ×d
a three-pixel wide narrowband (with 282,743 ×3 = 848,229 parameters). Now the
computational cost of a level-set formulation is not linear with parametric model.
It is a quadratic relationship!
In the general case, for N-D segmentation, let us assume that the number of
parameters used in parametric models is L, and AGF can gain some parameter
reduction factor of r in each dimension by using surface functions and spacing
of parametric nodal points of d-pixels in 1D, and narrowband width of levelsets is b (b ≥ 3) in each dimension, then the total number of parameters used in
each model is: (r)
(N−1)b(N−1)
d
−(N−1)
L for level-sets. In other words, if we use parametric models as the
reference method, and represent the computationalcomplexity in terms of the “BigOh” representation, then for N-D segmentation, AGF is a O(−(N −1)) method in
comparison with parametric models and level-sets with narrow-banding which is
O(N −1) method. The three models are only comparable when N = 2, in which
case N−1 = 1 and level-set with narrowband and parametric model are different by
a linear factor. However, AGF is still two orders of magnitude faster.
With this analysis, it is clearer of the advantage of AGF in comparison with
existing methods, especially for images with higher dimensionality. Nowadays, 3D
(N−1)
= a.
2
) grid points assuming
L for AGF, L for parametric model as the reference, and
232 Q. Duan et al.
and 4D imagedata are becoming routine in medicalimage analysis and we think that
the advantages for AGF in computationalefficiencyare critical and nonsubstitutable
by existing frameworks.
In general, deformable models usually utilize iterative methods to find the
optimal solution for the associated energy minimization framework via curve
evolution, which requires an additional variable as an artificial time step added into
the functions. In this case, curve evolution with explicit representation with K−node
points becomes an N ×K variable minimization problem since the evolving curve
is represented by
⎡
−→
0
X
−→
X
−→
X
1
.
.
.
K−1
(t)
(t)
(t)
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎤
⎥
⎥
⎥
=
⎥
⎥
⎦
⎢
⎢
⎢
⎢
⎢
⎣
x
0
x
0
1
x
0
K−1
0
(t) x
(t) x
.
.
.
(t) x
0
(t) ··· x
1
1
(t) ··· x
1
.
.
.
K−1
(t) ··· x
1
.
.
.
0
N−1
1
N−1
K−1
N−1
⎤
(t)
⎥
⎥
(t)
⎥
⎥
.
.
⎥
.
⎦
(t)
, (4)
with N ×K evolving variables.
Curve evolution with level-set becomes an (N + 1)-variate functional minimization problem since the evolving curve is represented by
φ
(−→X , t)=φ(x0, x1,...x
, t), (5)
N−1
which has to be solved for every point on the entire image domain or within the
narrowband.
Curve evolution with AGF becomes an N-variate functional minimization problem since the evolving curve can be represented by
(t)= f(x1, x2,...x
x
0
, t). (6)
N−1
The advantagein dimensionality reduction for surfacefunction actives over level-set
framework is evident.
The advantage of AGF over explicit expression is in two aspects. First, in explicit
representation, for each node point, there are N evolving variables, whereas in
surface function representation, there is only one variable for each corresponding
points. This will become more evident if we digitize (6) and reformulate in a similar
form as in (4) (Note in this case, AGF is operated at degenerated mode with reduced
efficiency compared with using basis functions):
⎡
−→
X
⎢
−→
⎢
X
⎢
⎢
⎢
⎣
−→
K−1
X
⎤
0
(t)
1
(t)
.
.
.
(t)
⎡
0
(t) x
x
⎢
⎢
⎢
⎢
⎢
⎣
x
0
1
x
0
K−1
0
(t) x
.
.
.
⎥
⎥
⎥
=
⎥
⎥
⎦
(t) x
0
1
1
1
.
.
.
K−1
1
··· x
··· x
.
.
.
··· x
⎤
0
N−1
⎥
1
⎥
N−1
⎥
. (7)
⎥
.
.
⎥
.
⎦
K−1
N−1
Real-Time 4D Cardiac Segmentation by Active Geometric Functions 233
Although the memory usage of (7)isthesameas(4), the curve evolution of (7)
has N − 1 less dimensionality than (4), which usually leads to faster and more
stable convergence. Generally speaking, the more parameters to be optimized, the
larger possibility that local minimums and saddle points exist, especially with
the presence of noise. Of course it is not necessarily true for every case that 1D
optimization is more stable than N-D; they could be equivalent. But even for that,
the searching space for 1D case is much smaller than the N-D one, which leads to
faster convergence.
Another aspect is that (6) can be represented via function basis, such as cubic
Hermite functions, in which case only a few weighting parameters rather than a lot
of digitized node points have to be stored and iterated on. This can further improve
the accuracy, efficiency, and numerical stability.
2.5 Beyond Efficiency Benefit
Beside the advantage brought by dimensionality reduction, AGF framework is also
benefited from its intrinsic function representation. By utilizing the idea of surface
function, rather than an explicit list or an implicit higher order function, AGF
can immediately utilizing some basic ideas in the algebra to further improve the
performance.
Basis representation is a very basic idea in function representation. By utilizing
surface function basisother than the nature Cartesian coordinates, AGF can not only
easily deal with enclosed shape as heart, liver, and various tumors but also easily
incorporate shape prior information. By utilizing function basis other than natural
basis, AGF can not only efficiently represent convoluted surfaces but also naturally
enforce prior knowledge on surface smoothness.
By using the concept of piecewise function, AGF can be extended with a
combination of finite element patches to capture much more complex shape, like
left ventricle. By incorporating repositioning and reorientation, the capture range of
AGF can be largely increased, giving less dependence on initialization.
2.6 Comparison with Other Deformable Models
Although as mentioned above, the interface functions for the three deformable
models are equivalent in terms of surface representation, different ways to approach
interface formulation provide different benefits and limitations.
Parametric active contours with explicit representations provide relative simple
representations through interface point coordinates and do not add additional
dimensionality to the optimization problem. However, it cannot easily handle
topological changes, and usually requires some prior knowledge about the target
topology for proper initialization. It is also not trivial to determine whether an
234 Q. Duan et al.
arbitrary pixel is inside or outside the segmented objects. Moreover, in order to
compare to other segmentation results such as manual tracing, it is usually not
very easy to directly compute quantitative metrics such as surface distances since
it requires pairing of closest points.
The level-set framework based on implicit representations via distance functions
can automatically deal with topological changes and allows easy determination
of whether a point is inside the object or not by simply looking at the sign of
the level-set function at the point location. However, the level-set formulation
implicitly introduces a new dimension, i.e., the value of the level-set function,
for each voxel in the whole image data space, whereas the other two models
only focus on the interface itself. This type of formulation implicitly increases
the dimensionality of the variational problem and thus increases the computational
cost of the optimization process. Even though a narrowband approach can improve
the efficiency by focusing only around the interface, it still requires more voxel
information than the other two formulations. In terms of segmentation comparison,
if the level-set function is the signed distance function, it is very easy to compute
the distance between surfaces, although in most of implementations, level-set
functions after few iterations do not necessarily remain as signed distance functions,
especially for those using narrowband approaches.
Active geometric function is a kind of marriage of the previously discussed
models: it focuses only on the interface as the explicit representation, while
being formulated as an implicit representation like the level-set framework. It has
advantage on dimensionality reduction in surface representation compared with
level-set. It can utilize function basis to avoid memory-inefficient boundary point
digitizing. Even if a degenerated digitized form has to be used and the surface
representation is similar to explicit expression, AGF still has been compared with
parametric deformable model. This dimensionality reduction gives AGF advantages
in efficiency in both aspects of the deformable model (i.e., surface modeling and
deformation scheme). Furthermore, with an implicit representation, it is straightforward to determine whether a point is inside the contour by simply comparing the
value of the surface function for that point with the value of the surface function
on the boundary. In addition, surface functions enable immediate quantitative
evaluation of the segmentation results via surface comparisons and differences in
surface function values. However, similar to parametric active contours, it is not
trivial to deal with topological changes, with its flexibility in topological changes
limited by the function basis used in the model.
3 Illustration on Synthetic Image
To illustrate the performance and someadvantages of the activegeometric functions,
several segmentation examples are presented in this section on a synthetic image.
This section specifically focuses on two implicit representation methods: the
AGF method and the level-set representation. Both segmentation frameworks use
Real-Time 4D Cardiac Segmentation by Active Geometric Functions 235
Fig. 2 (a) Synthetic image composed of two regions with normal distributions with the same mean
values but different standard deviations; (b) corresponding binary images indicating the ground
truth segmentation. The blue region has a standard deviation of 5 and the red one has a value of 10
in the original image. The interface is a sine function
variational formulae and interface functions. A fair head-to-head comparison is
possible by setting identical segmentation energy functional and numerical schemes
for both methods.
3.1 Synthetic Image
To illustrate the flexibility of the proposed AGF framework, instead of using an
example on common piecewise smooth images, in this section, both AGF and levelset approach were challenged with textured regions segmentation.
The synthetic image, as shown in Fig. 2a, was composed of two parts. Pixel
intensities for each part were randomly sampled from normal distributions with
identical mean values and different standard deviations. The corresponding groundtrue binary image is shown in Fig.2b. The blue region had a standard deviation of 5
and the red region had a value of 10. The interface between the regions was a sine
function. The dimension of the image was 65 by 65pixels.
3.2 AGF Using Numerical Solution
Usually in image segmentation, especially for the level-set framework, it is not
easy to find a closed form interface function. Instead, a numerical solution or
236 Q. Duan et al.
Fig. 3 (a) Initialization of the deformable model as in red and the ground truth interface in green;
(b) corresponding signed distance function for the level-set initialization
approximation of the interface function is computed via iterative numerical energy
minimization. With the level-set functions, this requires computation values of
the level-set function at each pixel (or on a narrowband near the interface). With
AGF, we only need to compute the surface function values at each pixel on the
interface.
Given the texture-based segmentation problem presented in Fig. 2, the following
energy functional was selected:
E =
(σ(x, y) −
Ω
δ
)2H(x, y)dxdy +
1
(σ(x, y) −
Ω
δ
)2(1 −H(x, y))dxdy, (8)
2
Ω
where
the image domain,σ(x, y) is a standard deviation estimator for pixel (x, y)
within a small neighborhood, and H is the Heaviside function,which equals 1 inside
the current interface and 0 outside. The parameters
δ
and
δ
1
are computed as the
2
average standard deviations inside and outside the current interface, respectively.
The optimal segmentation will partition the image into two regions, with relative
homogeneous distributions of the standard deviations within each region. This
approach is equivalent to segmenting a representation of local standard deviation
σ
(x, y) values of the image, knowing that for normal distributions N(μ,σ),average
standard deviations, converge to the scale parameter
σ
. The Chan–Vese level-set
numerical schemes described in [6] were used for the level-set implementation. For
simplification, no curvature constraints were used.
Both methods were initialized as a straight line at the center of the image, as
shown in Fig. 3a. The ground truth boundary is shown in green in the same figure.
Corresponding surface functions for AGF was just a 1 −D constant function as
y(x)=0, −32 ≤ x ≤ 32, whereas the corresponding level-set function was a plane
with slope 1 as shown in Fig. 3b.
Real-Time 4D Cardiac Segmentation by Active Geometric Functions 237
Fig. 4 Final segmentation (red line) compared to the ground truth (green line)for(a) active
geometric functions and (b) Chan–Vese level-set with identical segmentation energy functionals
and numerical schemes
Both methods were implemented in Matlabc. All computations were executed
on a 2.4 GHz 64-bit AMD server, running Red Hat Linux Enterprise AS. For each
pixel, a 7×7 neighborhood was used to compute the local standard deviations. An
artificial time step was set to five for both methods. The stability of the surface was
considered as the convergence criterion.
In order to quantitatively evaluate the segmentation result for each method, true
positive (TP) fraction ratio and false positive (FP) fraction ratio were computed.
In addition, root-mean squared error in distances from the interfaces to the ground
truth was estimated.
It took eight iterations (0.38 s) for the AGF to converge, each iteration taking
about 47.5 ms. The final segmentation result is shown in Fig. 4a, with red line
indicating the automatic segmentation and green line indicating the ground truth.
The TP fraction ration was 99.7%, whereas the FP ration was 3.8%. RMS error of
the distance to the ground truth was 1.56 pixels.
For comparison, it took 36 iterations (3.43 s) for the level-set approach to
converge, each iteration taking about 95.3ms. Final segmentation result is shown
in Fig. 4b, with the red line indicating the automatic segmentation and green line
indicating the ground truth. The TP fraction ration was 99.6%, whereas the false
positive fraction ration was 4.0%. The RMS error of the distance to the ground truth
was 2.15 pixels.
From this experiment, the AGF framework clearly demonstrates advantages in
computational efficiency when compared to the level-set framework, with not only
a shorter time per iteration but also fewer iterations. This is mainly due to the fact
that the level-set function hasto be updated overtheentire image domain whereas the
238 Q. Duan et al.
Fig. 5 (a) Initialization of the deformable model, in red, at left-most of the image with the
ground true interface in green; corresponding final segmentation (red line) compared to the ground
truth (green line)for(b) active geometric functions, and (c) Chan–Vese level-set under identical
segmentation energy functional and numerical schemes
AGF is only updated at the interface. Quantitative segmentation comparison yielded
comparable segmentationperformance forboth methods,AGF having slightly better
performance.
To test the ability of the proposed SFA framework under different and more
challenging initialization setups, another initialization on the left-most boundary
was tested as well for both methods. The initialization and final results from both
methods are shown in Fig.5.
Both methodsspent muchmore time to reachthe final results. It took25 iterations
for the AGF and 75 iterations for the level-set method to converge. Segmentation
by AGF yielded a TP of 99.81% and a FP of 4.03%. Segmentation by the levelset framework generated a TP of 99.91% and a FP of 6.58%. RMS errors were
1.61 pixels for AGF and 2.80 pixels for the level-set. Both methods had slightly
poorer performance compared with the results using closer initialization. However,
the AGF framework still exhibited advantages in efficiency and slightly better
performance.
It hasto be notedthat in numerical solution, AGF was operating in a “deenerated”
form which is similar to the parametric deformable model. But it still has the
advantage in efficiency over parametric deformable model since only one variable
per node is evolving, rather than N variables per node for the explicit representation.
3.3 AGF Using Analytical Solution or Function Basis
Another advantage of the AGF framework is to provide closed-form solution or
approximation for the interface, which, as indicated in the method section, can
bring additional gain in efficiency. Since the surface function is a (N −1) − D
function for N −D image data, we can choose arbitrary bases (naturalbases or other
bases) in the function space to express the surface function. Especially when some